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2.8 Basic Newton - Raphson (NR) Techniques

Before discussing the application of NR technique in load flow solution, let us first review the basicprocedure of solving a set of non-linear algebraic equation by means of NR algorithm. Let there be

n equations in n unknown variables x1, x2, xn as given below,

f1(x1, x2, xn) = b1f2(x1, x2, xn) = b2

fn

(x1, x2, xn

)= bn

(2.33)

In equation (2.33), the quantities b1, b2, bn as well as the functions f1, f2, fn are known.

To solve equation (2.33), first we take an initial guess of the solution and let these initial guesses be

denoted as, x(0)1

, x(0)2

, x(0)n . Subsequently, first order Taylors series expansion (neglecting the

higher order terms) is carried out for these equation around the initial guess of solution. Also let

the vector of initial guess be denoted as x(0) = x(0)1

, x(0)2

, x(0)n T. Now, application of Taylors

expansion on the equations of set (2.33) yields,

f1 x(0)1 , x(0)2 , x(0)n + f1x1x1 + f1

x2x2 + +

f1

xnxn = b1

f2 x(0)1 , x(0)2 , x(0)n + f2x1x1 +f2

x2x2 + +

f2

xnxn = b2

fn x(0)1 , x(0)2 , x(0)n + fnx1x1 +fn

x2x2 + +

fn

xnxn = bn

(2.34)

Equation (2.34) can be written as,

f1(x(0))f2(x(0))

fn(x(0))

+

f1

x1

f1

x2

f1

xn

f2

x1

f2

x2

f2

xn

fn

x1

fn

x2

fn

xn

x1

x2

xn

=

b1

b2

bn

(2.35)

In equation (2.35), the matrix containing the partial derivative terms is known as the Jacobin

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matrix (J). As can be seen, it is a square matrix. Hence, from equation (2.35),

x1

x2

xn

= [J]1

b1 f1(x(0))b2 f2(x(0))

bn fn(x(0))

= [J]1

m1

m2

mn

(2.36)

Equation (2.36) is the basic equation for solving the n algebraic equations given in equation

(2.33). The steps of solution are as follow:

Step 1: Assume a vector of initial guess x(0) and set iteration counter k = 0.

Step 2: Compute f1(x(k)), f2(x(k)), fn(x(k)).Step 3: Compute m1,m2, mn.

Step 4: Compute error =max [m1 , m2 , mn]Step 5: If error (pre - specified tolerance), then the final solution vector is x(k) and print

the results. Otherwise go to step 6.

Step 6: Form the Jacobin matrix analytically and evaluate it at x = x(k).

Step 7: Calculate the correction vector x = x1,x2, xnT by using equation (2.36).Step 8: Update the solution vector x(k+1) = x(k)+x and update k = k+1 and go back to step 2.

With this basic understanding of NR technique, we will now discuss the application of NR

technique for load flow solution. We will first discuss the Newton Raphson load- flow (NRLF) in

polar co-ordinates.

2.9 Newton Raphson load flow (NRLF) in polar co-ordinates

For NRLF techniques, the starting equations are same as those in equations (2.27) and (2.28), which

are reproduced below:

Pi =n

j=1

ViVjYij cos(i j ij) (2.37)Q

i=

n

j=1ViVj

Yijsin

(i

j

ij)

(2.38)

Now, as before let us again assume that in a n bus, m machine system, the first m buses

are the generator buses with bus 1 being the slack bus. Therefore, the unknown quantities are;

2, 3, n (total n-1 quantities) and Vm+1, Vm+2, Vn (total n-m quantities). Thus the total

number of unknown quantities is n 1+ n m = 2n m 1. Against these unknown quantities, the

specified quantities are; Psp

2, P

sp

3, P

spn (total n-1 quantities) and Q

sp

m+1, Qsp

m+2, Qspn (total

n-m quantities). Hence, the total number of specified quantities is also (2n m 1). Let thevectors of unknown quantities be denoted as =

[2, 3, n

]T

and V =

[Vm+1, Vm+2, Vn

]T

.

Similarly let the vector of the specified quantities be denoted as Psp = [Psp2 , Psp3 , Pspn ]T andQsp = [Qspm+1, Qspm+2, Qspn ]. Also note from equations (2.37) and (2.38) that the real and reactivepower injections at any bus are functions of and V. Thus, these injection quantities can be

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written as Pi = Pi(,V) for i = 2,3, n and Qi = Qi(,V) for i = (m + 1), (m + 2), n.For proceeding with NRLF, we assume initial guesses of the bus voltage angles (0) and the busvoltage magnitudes (V(0)). Subsequently, Taylors series expansion of equations (2.37) and (2.38)yields (following the same procedure as in the basic N-R technique),

P22

P2n

P2Vm+1

P2Vn

Pn

2

Pn

n

Pn

Vm+1

Pn

VnQm+1

2

Qm+1

n

Qm+1

Vm+1

Qm+1

Vn

Qn

2

Qn

n

Qn

Vm+1

Qn

Vn

2

n

Vm+1

Vm+1

=

Psp2

P2 (0),V(0)

Pspn Pn (0),V(0)

Qsp

m+1 Qm+1 (0),V(0)

Qspn Qn

(0),V(0)

(2.39)

In equation (2.39), the quantity Pi (0),V(0) is nothing but the calculated value of Pi withvectors (0),V(0). As a result, commonly, the quantity Pi (0),V(0) is denoted as Pcali . Withthese notations, equation (2.39) can be written as,

J1 J2J3 J4

V

= Psp PcalQsp Qcal

= PQ

(2.40)

In equation (2.40) the vectors Pcal and Qcal are defined as; Pcal = [Pcal2

, Pcal3

, Pcaln ]T andQcal = [Qcalm+1, Qcalm+2, Qcaln ]. Also note that the vectors and Psp are of dimension (n 1) 1each and the vectors V and Qsp are of dimension (n m)1 each. Therefore, from equations (2.39)and (2.40),

J1 =P

=

P2

2

P2

3

P2

n

P3

2

P3

3

P3

n

Pn

2

Pn

3

Pn

n

(2.41)

J2 =P

V=

P2

Vm+1

P2

Vm+2

P2

Vn

P3

Vm+1

P3

Vm+2

P3

Vn Pn

Vm+1

Pn

Vm+2

Pn

Vn

(2.42)

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J3 =Q

=

Qm+1

2

Qm+1

3

Qm+1

n

Qm+2

2

Qm+2

3

Qm+2

n

Qn2

Qn3

Qnn

(2.43)

J4 =Q

V=

Qm+1

Vm+1

Qm+1

Vm+2

Qm+1

Vn

Qm+2

Vm+1

Qm+2

Vm+2

Qm+2

Vn

Qn

Vm+1

Qn

Vm+2

Qn

Vn

(2.44)

In equations (2.41) to (2.44) the sizes of the various matrices are as follows: J1 (n1)(n1),J2 (n 1) (n m), J3 (n m) (n 1) and J4 (n m)(n m). Now, equation (2.40)can be written in compact form as,

[J] [X] = [M] (2.45)

In equation (2.45), the matrix J = J1 J2J3 J4

is known as the Jacobian matrix, the vector X=V

is known as the correction vector and the vector M = Psp PcalQsp Qcal

is known as themismatch vector. Further, the size of the matrix J is (2n m 1) (2n m 1) while the sizes ofboth the vectors X and M is (2n m 1) 1.

Equation (2.45) forms the basis of the NRLF (polar) algorithm, which is described below. Please

note that in the algorithm described below it is assumed that there is no generator which violates

its reactive power generation or absorption limit. The case of violation of reactive power generation

or absorption limit would be dealt with a little later.

Basic NRLF (polar) algorithm

Step 1: Initialise V(0)

j = Vsp

j 0o for j = 2,3, m and V

(0)j = 1.00

o for j = (m + 1), (m +2), n. Let the vectors of the initial voltage magnitudes and angles be denoted as V(0) and (0)respectively.

Step 2: Set iteration counter k = 1.

Step 3: Compute the vectors Pcal and Qcal with the vectors (k1) and V(k1) thereby forming

the vector M. Let this vector be represented as M = [M1,M2, M2nm1]T.Step 4: Compute error =max (M1 , M2 , M2nm1).Step 5: If error (pre - specified tolerance), then the final solution vectors are (

k1) and

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V(k1) and print the results. Otherwise go to step 6.

Step 6: Evaluate the Jacobian matrix with the vectors (k1) and V(k1).

Step 7: Compute the correction vector X by solving equation (2.45).

Step 8: Update the solution vectors (k) = (k1) + and V(k) = V(k1) + V. Update

k = k + 1 and go back to step 3.

In the above algorithm, the Jacobian matrix needs to be evaluated at each iteration. Therefore,

the element of the Jacobian matrix needs to be found out analytically. This is discussed next.

Formation of Jacobian matrix elements for NRLF (polar) technique

To derive the elements of the Jacobian matrix, let us revisit equations (2.37) and (2.38).

Pi =n

j=1

ViVjYij cos(i j ij) = V2i Gii + nj =1 i

ViVjYij cos(i j ij) (2.46)

Qi =n

j=1

ViVjYij sin(i j ij) = V2i Bii + nj =1 i

ViVjYij sin(i j ij) (2.47)

In the above two equations, the relations Gii=

Yii cos(ii) and Bii=

Yii sin(ii) have been used.From the expressions of the Pi and Qi in equations (2.46) and (2.47) respectively, the elements ofthe Jacobian matrix can be calculated as follows.

Matrix J1 = P

(in this case, i = 2,3, n, j = 2,3, n)Pi

j=

n

k=1 i

ViVkYik sin(i k ik); j = i (2.48)

Pi

j= ViVjYij sin(i j ij); j i (2.49)

Matrix J2 = PV

(in this case, i = 2,3, n, j = (m + 1), (m + 2), n)Pi

Vj= 2ViGii +

n

k=1 i

VkYik cos(i k ik); j = i (2.50)

Pi

Vj= ViYij cos(i j ij); j i (2.51)

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Matrix J3 = Q

(in this case, i = (m + 1), (m + 2), n, j = 2,3, n)Qi

j=

n

k =1 i

ViVkYik cos(i k ik); j = i (2.52)

Qi

j= ViVjYij cos(i j ij); j i (2.53)

Matrix J4 = QV

(in this case, i = (m + 1), (m + 2), n, j = (m + 1), (m + 2), n).Qi

Vj= 2ViBii +

n

k=1 i

VkYik sin(i k ik); j = i (2.54)

Qi

Vj= ViYij sin(i j ij); j i (2.55)

With these expressions of Jacobian elements given in equations (2.48)-(2.55), the Jacobian matrix

can be evaluated at each iteration as discussed earlier.

Now, in the basic NRLF (polar) algorithm described earlier, the generator Q-limits have not

been considered. To accommodate the generator Q-limits, at the beginning of each iteration, reac-

tive power absorbed or produced by each generator is calculated. If the calculated reactive power

is within the specified limits, the generator is retained as PV bus, otherwise the generator bus is

converted to a PQ bus, with the voltage at this bus no longer held at the specified value. The

detailed algorithm is as follows.

Complete NRLF (polar) algorithm

Step 1: Initialise V(0)

j = Vsp

j 0o for j = 2,3, m and V

(0)j = 1.00

o for j = (m + 1), (m +2), n. Let the vectors of the initial voltage magnitudes and angles be denoted as V(

0) and

(0)

respectively.

Step 2: Set iteration counter k = 1.

Step 3: For i = 2,3, m, carry out the following operations.

a) Calculate,

Q(k)i =

n

j=1

V(k1)i V

(k1)j Yij sin (k1)i (k1)j ij

b) If, Qmini Q(k)i Qmaxi ; then assign V(k)i = Vspeci and the ith bus is retained as PV busfor kth iteration.c) If Q

(k)i > Q

maxi , then assign Q

sp

i = Qmaxi or, if Q

(k)i < Q

min

i , then assign Qsp

i = Qmini . In

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both the cases, this bus is converted to PQ bus. Hence, its voltage magnitude becomes an unknown

for the present iteration (thereby introducing an extra unknown quantity) and to solve for this extra

unknown quantity, an extra equation is required, which is obtained by the new value of Qsp

i (as

shown above). Therefore, when the ith bus is converted to a PQ bus, the dimensions of both V

and Q vectors increases by one.

In general, ifl generator buses (l (m 1)) violate their corresponding reactive power limits atstep 3, then the dimensions of both V and Q vectors increases from (n m) to (n m + l).However, the dimensions of both P and vectors remain the same. Therefore, the size of

matrix J2 becomes (n 1) (n m + l), that of matrix J3 becomes (n m + l) (n 1) and thematrix J4 becomes of size (n m + l)(n m + l). The size of matrix J1, however, does not change.Hence, the size of the matrix J becomes (2n m 1 l) (2n m 1 l) while the sizes of boththe vectors X and M (in equation (2.45)) becomes (2n m 1 l) 1. Of course, if there isno generator reactive power limit violation, then l = 0.

Step 4: Compute the vectors Pcal and Qcal with the vectors (k1

) and V(k1) thereby formingthe vector M. Let this vector be represented as M = [M1,M2, M2nm1l]T.

Step 5: Compute error =max (M1 , M2 , M2nm1l).Step 6: If error (pre - specified tolerance), then the final rotation vectors are (

k1) and

V(k1) and print the results. Otherwise go to step 7.

Step 7: Evaluate the Jacobian matrix with the vectors (k1) and V(k1).

Step 8: Compute the correction vector X by solving equation (2.45).

Step 9: Update the solution vectors (k) = (k1) + and V(k) = V(k1) + V. Update

k = k + 1 and go back to step 3.

In the next lecture, we will look at an example of NRLF (polar) technique.

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